How to create automatic follow-up (delayed) animation for child objects?
You can use the Slow Parent Option available in the object relations properties.



After setting up you parent/child chain of objects go to each object individually and under the Properties Window > Object > Relations Extras tick Slow Parent option. Bellow it set the desired delay in frames.



You can select all relevant objects and click the properties wile pressing the Alt key to set the values for several selected objects at once. Alternatively right click and choose Copy to Selected.